Tight Sum-of-Squares lower bounds for binary polynomial optimization problems-Reference-Cited by-同舟云学术

Tight Sum-of-Squares lower bounds for binary polynomial optimization problems

Published:2023-10-17 Issue: Volume: Page:
ISSN:1942-3454
Container-title:ACM Transactions on Computation Theory
language:en
Short-container-title:ACM Trans. Comput. Theory

Author:

Kurpisz Adam¹,Leppänen Samuli²,Mastrolilli Monaldo³

Affiliation:

1. Business School, Bern University of Applied Sciences, Switzerland and ETH Zürich, Department of Mathematics, Switzerland

2. Axpo Solutions AG, Switzerland

3. SUPSI-IDSIA - Dalle Molle Institute for Artificial Intelligence, Switzerland

Abstract

For binary polynomial optimization problems of degree 2 d with n variables Sakaue, Takeda, Kim and Ito [SIAM J. Optim., 2017] proved that the

\(\lceil \frac{n+2d-1}{2}\rceil \)

th semidefinite (SDP) relaxation in the SoS/Lasserre hierarchy of SDP relaxations provides the exact optimal value. When n is an odd number, we show that their analysis is tight, i.e. we prove that

\(\frac{n+2d-1}{2} \)

levels of the SoS/Lasserre hierarchy are also necessary. Laurent [Math. Oper. Res., 2003] showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull of a linear representation of a set with no integral points. She conjectured that the SoS/Lasserre rank for the same problem is n − 1. In this paper we disprove this conjecture and derive lower and upper bounds for the rank.

Publisher

Association for Computing Machinery (ACM)

Subject

Computational Theory and Mathematics,Theoretical Computer Science

Link

https://dl.acm.org/doi/pdf/10.1145/3626106

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